Delicate Topology, Part I

Pontrjagin's seminal topological classification of two-band Hamiltonians in three momentum dimensions is hereby enriched with the inclusion of a crystallographic rotational symmetry. The enrichment is attributed to a new topological invariant which quantifies a 2pi-quantized change in the Berry-Zak phase between a pair of rotation-invariant lines in the Brillouin zone, henceforth referred to as a returning Thouless pump (RTP). We show that the RTP is associated with metallic in-gap states under open boundary conditions with sharply-terminated hoppings; more generally, the RTP is associated to anomalous fractional Berry-Zak phases of surface states, no matter how the hoppings are terminated. The RTP adds to the family of topological invariants (the Hopf and Chern numbers) that are known to classify two-band Hamiltonians in Wigner-Dyson symmetry class A. Of these, the RTP and Hopf invariants are delicate, meaning that they can be trivialized by adding a particular trivial band to either the valence or the conduction subspace. Not all trivial band additions will nullify the RTP invariant, which allows its generalization beyond two-band Hamiltonians to arbitrarily many bands; such generalization is a hallmark of symmetry-protected delicate topology.